Finding Critical Values. In Exercises 5–8, find the critical value $z_{\frac{\alpha }{2}}$that corresponds to the given confidence level.

98%

The level of significance is 98%.

A critical value is a point on the test distribution that is compared to the test statistics to determine whether to reject the null hypothesis. It is denoted by $z_{\frac{\alpha }{2}}$which is equal to z score within the area of $\frac{\alpha }{2}$in the right tail of the standard normal distribution for $\alpha$level of significance.

When finding a critical value $z_{\frac{\alpha }{2}}$for a particular value of $\alpha$, note that$\frac{\alpha }{2}$is the cumulative area to the right of $z_{\frac{\alpha }{2}}$which implies that the cumulative area to the left of $z_{\frac{\alpha }{2}}$must be.

Here, for 98% confidence level,

$$\begin{gathered} \alpha =0.02 \\ 1-\frac{\alpha }{2}=0.99 \end{gathered}$$

To find the z score corresponding to the area 0.9900,

In the standard normal table for positive z score, find the value closest to 0.9900, which is 0.9901, corresponding row value is 2.3 and column values is 0.03which corresponds to the z-score of 2.33.

Therefore, the critical value for 98% level of significance is 2.33.